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A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.

%I A338911 #24 Jul 27 2025 08:57:41
%S A338911 9,21,39,49,57,87,91,111,129,133,159,169,183,203,213,237,247,259,267,
%T A338911 301,303,321,339,361,371,377,393,417,427,453,481,489,497,519,543,551,
%U A338911 553,559,579,597,623,669,687,689,703,707,717,749,753,789,791,793,813,817
%N A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.
%F A338911 Numbers m such that A001222(m) = 2 and A195017(m) = -2. - _Peter Munn_, Jan 17 2021
%e A338911 The sequence of terms together with their prime indices begins:
%e A338911       9: {2,2}     237: {2,22}    481: {6,12}
%e A338911      21: {2,4}     247: {6,8}     489: {2,38}
%e A338911      39: {2,6}     259: {4,12}    497: {4,20}
%e A338911      49: {4,4}     267: {2,24}    519: {2,40}
%e A338911      57: {2,8}     301: {4,14}    543: {2,42}
%e A338911      87: {2,10}    303: {2,26}    551: {8,10}
%e A338911      91: {4,6}     321: {2,28}    553: {4,22}
%e A338911     111: {2,12}    339: {2,30}    559: {6,14}
%e A338911     129: {2,14}    361: {8,8}     579: {2,44}
%e A338911     133: {4,8}     371: {4,16}    597: {2,46}
%e A338911     159: {2,16}    377: {6,10}    623: {4,24}
%e A338911     169: {6,6}     393: {2,32}    669: {2,48}
%e A338911     183: {2,18}    417: {2,34}    687: {2,50}
%e A338911     203: {4,10}    427: {4,18}    689: {6,16}
%e A338911     213: {2,20}    453: {2,36}    703: {8,12}
%p A338911 q:= n-> (l-> add(i[2], i=l)=2 and andmap(i->
%p A338911     numtheory[pi](i[1])::even, l))(ifactors(n)[2]):
%p A338911 select(q, [$1..1000])[];  # _Alois P. Heinz_, Nov 23 2020
%t A338911 Select[Range[100],PrimeOmega[#]==2&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]
%o A338911 (Python)
%o A338911 from math import isqrt
%o A338911 from sympy import primerange, primepi
%o A338911 def A338911(n):
%o A338911     def bisection(f,kmin=0,kmax=1):
%o A338911         while f(kmax) > kmax: kmax <<= 1
%o A338911         kmin = kmax >> 1
%o A338911         while kmax-kmin > 1:
%o A338911             kmid = kmax+kmin>>1
%o A338911             if f(kmid) <= kmid:
%o A338911                 kmax = kmid
%o A338911             else:
%o A338911                 kmin = kmid
%o A338911         return kmax
%o A338911     def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),-1) if a&1^1)
%o A338911     return bisection(f,n,n) # _Chai Wah Wu_, Apr 03 2025
%Y A338911 A338910 is the odd instead of even version.
%Y A338911 A339004 is the squarefree case.
%Y A338911 A001221 counts distinct prime indices.
%Y A338911 A001222 counts prime indices.
%Y A338911 A001358 lists semiprimes, with odd/even terms A046315/A100484.
%Y A338911 A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
%Y A338911 A338899, A270650, A270652 list prime indices of squarefree semiprimes.
%Y A338911 A289182/A115392 list the positions of odd/even terms of A001358.
%Y A338911 A300912 lists semiprimes with relatively prime indices.
%Y A338911 A318990 lists semiprimes with divisible indices.
%Y A338911 A338904 groups semiprimes by weight.
%Y A338911 A338906/A338907 list semiprimes of even/odd weight.
%Y A338911 A338909 lists semiprimes with non-relatively prime indices.
%Y A338911 A338912 and A338913 list prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
%Y A338911 Cf. A005117, A037143, A055684, A056239, A065516, A112798, A128301, A195017, A320655, A320732, A320892, A338898, A339002, A339003.
%K A338911 nonn
%O A338911 1,1
%A A338911 _Gus Wiseman_, Nov 20 2020